Light neutrinos, Dark matter and leptogenesis near electroweak scale and $Z_4$ symmetry

This paper proposes a Type I seesaw model with $Z_4$ symmetry and soft-breaking terms at the electroweak scale that successfully explains light neutrino oscillation data, identifies the lightest right-handed neutrino as a dark matter candidate via freeze-in, and generates the baryon asymmetry through resonant leptogenesis using only a minimal set of parameters.

The Gist

Imagine the universe as a giant, complex puzzle. For a long time, scientists have been trying to fit three specific pieces together:

1. Why do neutrinos (tiny ghost particles) have mass?
2. What is Dark Matter (the invisible stuff holding galaxies together)?
3. Why is there more matter than antimatter in the universe? (Otherwise, they would have cancelled each other out, and we wouldn't be here).

Usually, solving these puzzles requires adding heavy, complicated machinery to our theories, pushing the "new physics" to energy scales so high we can never test them.

This paper proposes a much more elegant solution. The authors, Kunal Pandey and Rathin Adhikari, suggest that if we introduce a specific set of rules called $Z_4$ symmetry (think of it as a strict "dance choreography" that particles must follow), we can solve all three mysteries using a scale of energy that is actually reachable by our current or near-future particle colliders.

Here is the breakdown of their solution using simple analogies:

1. The "Ghostly" Neutrinos and the Seesaw

In the standard model of physics, neutrinos are like ghosts; they barely interact with anything. The "Type-I Seesaw" mechanism is a popular theory explaining why they are so light. Imagine a seesaw: on one end is a heavy, invisible weight (a heavy Right-Handed Neutrino), and on the other is a tiny, visible weight (our light neutrino). The heavier the invisible weight, the lighter the visible one.

Usually, to get our neutrinos to be as light as they are, that invisible weight needs to be astronomically heavy (like a mountain). But this paper suggests a trick. By applying the $Z_4$ symmetry, the authors arrange the "seesaw" so that at the very beginning (the "tree level"), the light neutrinos are perfectly massless.

The Analogy: Imagine a perfectly balanced scale where the light side is weightless. This isn't a mistake; it's a feature of the symmetry.

2. Waking Up the Neutrinos (One-Loop Corrections)

If the neutrinos are massless, how do we get the tiny masses we observe? The authors say: "Let's add a little bit of noise."

In quantum physics, particles are constantly interacting with a "soup" of other particles. These interactions create tiny ripples. The authors calculate these ripples (called one-loop corrections).

• The Result: These tiny ripples break the perfect balance of the massless scale. The neutrinos gain a tiny, non-zero mass.
• The Magic: Because the starting point was so perfectly symmetrical, these tiny ripples are enough to explain the exact mass differences and mixing angles we see in experiments, without needing to fine-tune the numbers manually. It's like a perfectly balanced mobile that only needs a gentle breeze to start moving in the exact pattern we want.

3. The Dark Matter Candidate (The "Feeble" Neighbor)

The model introduces three heavy neutrinos. Two of them are heavy and interact normally, but the third one ($N_1$) is special.

• The Analogy: Imagine a party where everyone is dancing and talking (thermal equilibrium). The third heavy neutrino is like a shy guest who barely speaks to anyone. Because of the soft symmetry breaking (a tiny crack in the perfect dance rules), this guest interacts so weakly that they never join the party.
• The Freeze-In Mechanism: Instead of being "frozen out" (leaving the party early), this shy guest is slowly "frozen in." They are produced very slowly over time by the decay of other particles. By the time the universe cools down, this shy guest has accumulated just the right amount to be the Dark Matter we observe.

4. Creating the Universe's Matter (Resonant Leptogenesis)

How did we get more matter than antimatter? This is called Leptogenesis.

• The Setup: The two heavier neutrinos ($N_2$ and $N_3$) are almost identical twins (quasi-degenerate). They are so close in mass that they are like two tuning forks vibrating at nearly the same frequency.
• The Resonance: When these twins decay, their near-identical masses cause a "resonance" (like pushing a swing at just the right moment). This amplifies a tiny difference between matter and antimatter creation.
• The Outcome: This process creates a slight excess of leptons (electrons/neutrinos), which the universe's "machinery" (sphalerons) converts into an excess of protons and neutrons. This explains why we have a universe made of stuff, not just empty space.

5. Why This is a Big Deal

• Low Energy Scale: Most theories say these heavy particles are too heavy to ever find (like looking for a needle in a galaxy-sized haystack). This paper says they are around 152 GeV. This is the energy scale of the Large Hadron Collider (LHC). It's like saying the needle is actually in the garden, not the galaxy.
• Minimal Parameters: They managed to solve all three problems (Neutrino mass, Dark Matter, Matter/Antimatter asymmetry) using very few adjustable numbers. It's a "lean" solution.
• Testability: Because the heavy neutrinos are light enough, we might be able to detect them in future experiments, specifically at electron-proton colliders or by looking for very specific decay patterns at the LHC.

Summary

The authors built a house of cards using a strict set of rules ($Z_4$ symmetry).

1. The rules make the neutrinos massless at first.
2. Tiny quantum ripples give them the exact mass we see.
3. One heavy neutrino is too shy to interact, becoming Dark Matter.
4. Two other heavy neutrinos are twins that resonate to create the matter in our universe.
5. All of this happens at an energy level we can actually test, making this a very exciting and testable theory.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain the origin of non-zero neutrino masses and the observed baryon asymmetry of the universe (BAU). The canonical Type-I seesaw mechanism introduces three heavy Right-Handed Neutrinos (RHNs) but typically requires a high mass scale ($\sim 10^9$ GeV) to generate light neutrino masses ($\sim 0.1$ eV) with order-unity Yukawa couplings. This high scale makes experimental verification difficult.

The authors aim to address three simultaneous challenges within a low-scale Type-I seesaw scenario (near the electroweak scale, $\sim$ TeV):

1. Neutrino Oscillations: Reproducing light neutrino mass-squared differences, mixing angles, and CP-violating phases within $3\sigma$ experimental limits.
2. Dark Matter (DM): Identifying a viable DM candidate.
3. Baryogenesis: Explaining the BAU via Resonant Leptogenesis (RL).

A key challenge in low-scale seesaw models is that large Yukawa couplings are often required, which can conflict with experimental bounds or require fine-tuning to keep light neutrinos massless at tree level before radiative corrections.

2. Methodology

The authors propose a specific model extension of the SM involving:

• Field Content: Three heavy Majorana RHNs ($N_1, N_2, N_3$) and the standard SM fields.
• Symmetry: A discrete $Z_4$ symmetry is imposed on the Lagrangian. Unlike previous works where $Z_4$ charges were linked to Baryon/Lepton numbers, here specific charges are assigned to different RHNs to naturally generate a specific texture for the mass matrices without fine-tuning.
• Massless Texture: The $Z_4$ symmetry enforces a "massless texture" where all three light neutrinos are massless at the tree level ($M_{\nu}^{tree} = 0$) despite having non-vanishing Dirac mass matrices ($M_D$).
• Radiative Corrections: Light neutrino masses are generated via one-loop corrections to the seesaw mass matrix. The authors calculate corrections to both the Dirac ($M_D$) and Majorana ($M_L$) blocks.
• Soft Symmetry Breaking: Small soft-breaking terms ($M_3, M_5, k_1$) are introduced to:
  • Break the exact degeneracy of heavy neutrinos for Resonant Leptogenesis.
  • Generate tiny couplings for the lightest RHN to act as Dark Matter via the freeze-in mechanism.
  • Provide the necessary CP-violating phases.

3. Key Contributions

A. Theoretical Framework and $Z_4$ Symmetry

The authors construct a specific texture for the Dirac ($M_D$) and Majorana ($M_R$) mass matrices.

• $M_D$ Structure: The $Z_4$ symmetry forces the first two columns of $M_D$ to vanish at tree level, leaving only the third column non-zero.
• $M_R$ Structure: The symmetry allows specific mass terms ($M_1, M_6$) while forbidding others ($M_3, M_5$) at the tree level.
• No Fine-Tuning: This structure naturally results in three massless light neutrinos at tree level without requiring fine-tuned relations between Yukawa couplings and RHN masses.

B. One-Loop Mass Generation

The paper demonstrates that in this specific texture:

• The one-loop corrections to the $M_L$ block (usually dominant in general seesaw models) vanish due to the specific structure of the diagonalized $M_R$ and the zero first column of $M_D$.
• Consequently, the one-loop corrections to the $M_D$ block become the dominant source of light neutrino masses.
• This mechanism allows for light neutrino masses ($\sim 0.1$ eV) with RHN masses around the TeV scale and relatively large Yukawa couplings.

C. Dark Matter Candidate (Freeze-in)

• The lightest RHN ($N_1$) is decoupled from SM interactions at tree level.
• A soft symmetry-breaking term ($M_5 \sim 10^{-10}$ GeV) induces a tiny mixing between $N_1$ and the active neutrinos.
• This results in extremely small Yukawa couplings ($\sim 10^{-14}$), making $N_1$ a Feebly Interacting Massive Particle (FIMP).
• $N_1$ is produced via the freeze-in mechanism (decays of $W, Z, h$ bosons) and never reaches thermal equilibrium, naturally explaining the observed relic abundance.

D. Resonant Leptogenesis (RL)

• The two heavier RHNs ($N_2, N_3$) are quasi-degenerate in mass ($M_2 \approx M_3$).
• A small soft-breaking term ($M_3$) breaks this degeneracy slightly, satisfying the resonance condition ($M_3 - M_2 \sim \Gamma/2$).
• Another soft-breaking term ($k_1$) introduces a small imaginary part to the Yukawa couplings, generating the necessary CP asymmetry.
• The authors account for flavor effects and thermal corrections (effective Higgs mass) in the early universe, which are crucial at the electroweak scale.

4. Results

• Neutrino Oscillations: The model successfully reproduces the observed neutrino oscillation data (mass-squared differences $\Delta m^2_{21}, \Delta m^2_{31}$, mixing angles $\theta_{12}, \theta_{23}, \theta_{13}$, and CP phase $\delta_{CP}$) within the $3\sigma$ confidence level.
  • Parameters: Only 3 complex Yukawa parameters ($k, \alpha, \beta$) and 1 real mass parameter ($M_6$) are required to fit the oscillation data.
  • Benchmark Points: Four benchmark points (BP1-BP4) are provided. For BP1, $M_6 \approx 152$ GeV.
• Dark Matter:
  • The model predicts a lightest RHN mass ($M_1$) in the range $10^{-6} \text{ GeV} \lesssim M_1 \lesssim 5 \times 10^{-4} \text{ GeV}$.
  • The required soft-breaking parameter $M_5$ is constrained to $1.78 \times 10^{-11} \text{ GeV} \lesssim M_5 \lesssim 4.15 \times 10^{-10} \text{ GeV}$ to match Planck relic density data.
  • The sum of light neutrino masses is $\sum m_i \approx 0.058$ eV, consistent with cosmological bounds.
• Baryon Asymmetry:
  • Resonant Leptogenesis successfully generates the observed baryon asymmetry ($Y_{\Delta B} \approx 8.7 \times 10^{-11}$) for RHN masses around 152 GeV.
  • The inclusion of the thermal Higgs mass is critical; it suppresses the decay width, reducing the washout effect and allowing the asymmetry to survive.
• Experimental Constraints:
  • The heavy-light mixing angles ($|U_{\nu N}|^2$) for $N_2, N_3$ are calculated to be $\sim 10^{-7} - 10^{-6}$.
  • These values are well below current LHC limits ($\sim 10^{-2}$) but are potentially detectable in future electron-proton colliders (e.g., LHeC or FCC-eh) which have lower QCD backgrounds.

5. Significance

This work presents a highly economical and testable framework for physics beyond the Standard Model:

1. Low Scale Accessibility: It pushes the seesaw scale down to the electroweak scale ($\sim 150$ GeV), making the heavy neutrinos potentially accessible to future colliders, unlike the traditional high-scale seesaw.
2. Unified Solution: It simultaneously explains neutrino masses, Dark Matter, and Baryogenesis using a minimal set of parameters and a single symmetry principle ($Z_4$).
3. Naturalness: The "massless texture" at tree level is achieved without fine-tuning, relying instead on the discrete symmetry.
4. Radiative Dominance: It highlights a unique scenario where loop corrections to the Dirac mass matrix dominate over the Majorana mass matrix corrections, a feature specific to the proposed $Z_4$ texture.
5. Predictive Power: The model makes specific predictions for the mass of the lightest RHN (keV-MeV scale for DM) and the mixing angles of the heavier RHNs, guiding future experimental searches.